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In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. [1]
Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation + = if n is an integer greater than two (n > 2).. Over time, this simple assertion became one of the most famous unproved claims in mathematics.
Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation a n + b n = c n for any integer value of n greater than 2. (For n equal to 1, the equation is a linear equation and has a solution for every possible a and b.
Fermat's last theorem Fermat's last theorem, one of the most famous and difficult to prove theorems in number theory, states that for any integer n > 2, the equation a n + b n = c n has no positive integer solutions. Fermat's little theorem Fermat's little theorem field extension A field extension L/K is a pair of fields K and L such that K is ...
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer solutions to a n + b n = c n; Fermat's little theorem, a property of prime numbers; Fermat's theorem on sums of two squares, about primes expressible as a sum of ...
Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either (p, p − 3) or (p, p − 5) fails to be an irregular pair. ((p, 2k) is an irregular pair when p is irregular due to a certain condition described below being realized at 2k.)
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
While the auxiliary prime has nothing to do with the divisibility by and must also divide either , or for which the violation of the Fermat Theorem would occur and most likely the conjecture is true that for given the auxiliary prime may be arbitrarily large similarly to the Mersenne primes she most likely proved the theorem in the general case by her considerations by infinite ascent because ...